## Roadmap for Formal Mathematical Physics Content

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Suppose an object starts an infinite distance from a moon and is dropped, falling towards the moon due to gravitational acceleration. What is the speed of the object when it is distance $$r$$ from the moon?

Figure 1: small mass falling towards a moon from initial position at infinity.

The initial condition is $$v(x=\infty) =0 \label{eq:initial_velocity}$$ The force acting on the object is $$\vec{F} = \frac{-G m_1 m_2}{x^2} \hat{x} \label{eq:gravitational force}$$ The work is calculated using W = $$\Delta E$$ since the force changes. To find the cumulative work done on the object, integrate over all positions between $$\infty$$ and $$r$$ $$W = \int_{\infty}^r \vec{F}\cdot d\vec{r} \label{eq:work as function of force}$$ Substituting the gravitational force into Eq. \ref{eq:work as function of force}, $$W = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx$$ Factor out the constants, $$W = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx$$ which leads to $$W = \frac{G m_1 m_2}{r}$$ Another definition of work is that it is the change in energy for a system: $$W = \Delta KE$$. Because the initial velocity was zero, the work here is $$W = \frac{1}{2} m_1 v^2$$ Thus we can combine the two definitions of work to get $$W = \frac{1}{2} m_1 v^2 = \frac{G m_1 m_2}{r}$$ The $$m_1$$ cancels, leaving $$v(r) = \sqrt{\frac{2Gm_2}{r}}$$